Subtract the following rational expressions. $\dfrac{3m^4}{3m+2}-\dfrac{4m^2}{5m+2}=$
Answer: We can subtract two rational expressions whose denominators are equal by subtracting the numerators and keeping the denominator the same. [Does this fit with how we subtract rational numbers?] When the denominators are not the same, we must manipulate them so that they become the same. In other words, we must find a common denominator. Since the two denominators do not share any common factors, the common denominator is simply the product of these two denominators: $({3m+2})\cdot({5m+2})$. Let's manipulate the expressions to have that denominator: $\begin{aligned} &\phantom{=}\dfrac{3m^4}{{3m+2}}-\dfrac{4m^2}{{5m+2}} \\\\ &=\dfrac{3m^4\cdot({5m+2})}{({3m+2})\cdot({5m+2})}-\dfrac{4m^2\cdot({3m+2})}{({5m+2})\cdot({3m+2})} \end{aligned}$ [Why did we do that?] Now that both denominators are the same, let's subtract! $\begin{aligned} &\phantom{=}\dfrac{3m^4\cdot(5m+2)}{(3m+2)\cdot(5m+2)}-\dfrac{4m^2\cdot(3m+2)}{(5m+2)\cdot(3m+2)} \\\\ &=\dfrac{3m^4\cdot(5m+2)-4m^2\cdot(3m+2)}{(3m+2)(5m+2)} \\\\ &=\dfrac{15m^5+6m^4-12m^3-8m^2}{(3m+2)(5m+2)} \end{aligned}$ In conclusion, $\dfrac{3m^4}{3m+2}-\dfrac{4m^2}{5m+2}=\dfrac{15m^5+6m^4-12m^3-8m^2}{(3m+2)(5m+2)}$